What do you seek to achieve with this complaint?
Is that goal realistic/achievable?
Have your NHS Trust done everything they can to try and reschedule your care?
What mitigating circumstance do they have and have their delays been unavoidable?
Is your complaint likely to be successful?
Normally, I'd encourage complaints if you feel your care has been poor, but it doesn't sound like you have been treated unfairly or badly - just delayed. And it doesn't sound like its life threatening, thus is lower priority whilst they clear more serious issues. That makes me question whether it would change anything or improve your situation and might just give you additional stress.
I sympathetise, but the point of complaints is in response to poor care / to get change.
In this situation as much as it sucks, you are kind of stuffed as I doubt there is very much that can be done to resolve the situation but to wait or go private if you can.
Ultimately, your issues are not the most serious and there aren't enough staff and thats the bottom line. If they push you forward that has knock on effects on other patients. And they have a duty of care to them too. So they have no choice but to prioritise on the basis of need - and that means life threatening conditions come first not fertility treatment.
Being pushy isn't going to change the situation under the circumstances. There simply are too many other people in similar or worse situations.
Making a complaint isn't going to make you feel better or help anyone sadly.